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- W1756132046 abstract "We investigate symmetric, suitably normalized orthogonal polynomial sequences $(P_n(x))_{nin mathbb {N}_0}$ and characterize the class of ultraspherical polynomials in terms of certain constancy properties of the Fourier coefficients which belong to $(P_{2n-1}^prime (x))_{nin mathbb {N}}$. Similar characterizations are obtained for the discrete, resp. continuous, $q$-ultraspherical polynomials after replacing the derivative $frac {mathrm {d}}{mathrm {d}x}$ by the $q$-difference operator $D_{q^{-1}}$, resp. AskeyâWilson operator $mathcal {D}_q$. Our results extend earlier work of LasserâObermaier and IsmailâObermaier where the whole sequences $(P_n^prime (x))_{nin mathbb {N}}$, $(D_{q^{-1}}P_n(x))_{nin mathbb {N}}$ and $(mathcal {D}_q P_n(x))_{nin mathbb {N}}$ had to be taken into account; we shall see that the characterizing properties concerning the even indices turn out to be redundant. We also characterize a large subclass of the continuous $q$-ultraspherical polynomials via the averaging operator $mathcal {A}_q$, and we show that this characterization cannot be extended to the whole class." @default.
- W1756132046 created "2016-06-24" @default.
- W1756132046 creator A5081706046 @default.
- W1756132046 date "2015-09-04" @default.
- W1756132046 modified "2023-09-27" @default.
- W1756132046 title "Characterizations of ultraspherical polynomials and their $q$-analogues" @default.
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- W1756132046 doi "https://doi.org/10.1090/proc/12640" @default.
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